The solution of (x+ y) d y+y d x=0 when y(0)=1 is

Step-by-step solution

The given differential equation is (x + y) dy + y dx = 0 . Rearranging the terms, we get y dx + (x + y) dy = 0 , which can be written as y dx + x dy + y dy = 0 . This is equivalent to d(xy) + y dy = 0 . Integrating both sides, we have d(xy) + y dy = C . Using integration by parts for y dy , we get y y - y 1 y dy = y y - y . Thus, the general solution is xy + y y - y = C . Given the initial condition y(0) = 1 , we substitute x = 0 and y = 1 into the general solution: 0(1) + 1 (1) - 1 = C 0 + 0 - 1 = C C = -1 . Subst